#9267007

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3770 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,5)(3,9,11,7)(4,8,12,13)(10,15,16,14)(34,40)(35,43)(36,46)(38,48);(2,5)(7,9)(8,13)(14,15)(17,18,23,25)(19,26,20,21)(22,30,31,24)(27,29,28,32)(33,34,39,40)(35,41,43,37)(36,42,46,45)(38,47,48,44);(4,12)(8,13)(10,16)(14,15)(17,19,23,20)(18,21,25,26)(22,27,31,28)(24,29,30,32)(33,35)(34,41)(36,47)(37,40)(38,45)(39,43)(42,48)(44,46);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,16)(13,15)(17,20,23,19)(18,21,25,26)(22,28,31,27)(24,29,30,32)(33,36)(34,42)(35,44)(37,48)(38,41)(39,46)(40,45)(43,47);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,21)(18,19)(20,25)(22,29)(23,26)(24,27)(28,30)(31,32)(33,37,39,41)(34,43,40,35)(36,48,46,38)(42,47,45,44);(1,5,6,2)(3,9,11,7)(4,13,12,8)(10,15,16,14)(17,22)(18,24)(19,27)(20,28)(21,29)(23,31)(25,30)(26,32)(33,38,39,48)(34,44,40,47)(35,45,43,42)(36,41,46,37)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=i,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9174203);